Fibonacci Iterations.

There exist unique positive integers $n_1, n_2, n_3, n_4, n_5, n_6$ such that

$\large\ \displaystyle \sum _{ { i }_{ 1 } = 0 }^{ 100 }{ \sum _{ { i }_{ 2 } = 0 }^{ 100 }{ \sum _{ { i }_{ 3 } = 0 }^{ 100 }{ \sum _{ { i }_{ 4 } = 0 }^{ 100 }{ \sum _{ { i }_{ 5 } = 0 }^{ 100 }{ { F }_{ { i }_{ 1 } + { i }_{ 2 } + { i }_{ 3 } + { i }_{ 4 } + { i }_{ 5 } } } } } } } = { F }_{ { n }_{ 1 } } - 5{ F }_{ { n }_{ 2 } } + 10{ F }_{ { n }_{ 3 } } - 10{ F }_{ { n }_{ 4 } } + 5{ F }_{ { n }_{ 5 } } - { F }_{ { n }_{ 6 } }$

where $F_n$ is the Fibonacci Number with $F_0 = 0, F_1 = 1$ .

Find $n_1 + n_2 + n_3 + n_4 + n_5 + n_6$ .